Variational Bayesian Inference for Soft Sensor Development With Irregular Measurements

doi:10.1109/tim.2026.3712920

Variational Bayesian Inference for Soft Sensor Development With Irregular Measurements

2026-07-13

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DOI：10.1109/tim.2026.3712920

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Many challenges still exist in developing soft sensors. Due to the influence of various factors, the collected industrial process data are inevitably contaminated by measurement noise, outliers, and missing data. This article proposes a robust probabilistic modeling method for soft sensor development and its extended version for multirate systems based on variational Bayesian inference (VBI). The Laplace distribution is used to model measurement noise and outliers because of its heavy-tailed nature. All unmeasurable variables in multirate systems are treated as hidden variables. The VBI method is utilized to simultaneously estimate these hidden variables and unknown parameters. Additionally, the case of missing output data is considered and addressed within the VBI framework. A numerical example, a continuous fermentation reactor process, and a pilot-scale hybrid tank experiment are utilized to validate the robustness and effectiveness of the proposed methods. Simulation results demonstrate that the proposed methods exhibit robust performance in simultaneously estimating time delays and handling irregular measurement data.

Keywords:

Irregular measurement data

Laplace distribution

multirate systems

soft sensor

variational Bayesian inference (VBI)

Variational Bayesian Inference for Soft Sensor Development With Irregular Measurements

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